Chapter 5

Measures of Dispersion

What is Dispersion as a Concept?

image:geeksforgeeks

What is Dispersion within Statistics?

It is the amount of spread or variability among raw scores in a distribution.

image:geeksforgeeks

Kurtosis - the Cousin of Skewness

The ‘sharpness’ of the peak of a frequency-distribution curve.


The ‘sharpness’ of the peak occurs due to the tailedness (i.e., how often outliers occur) with the distribution.

image:Dynamics

Types of Kurtosis

Leptokurtosis

data clustered around the mean and fewer data points in the tails compared to a normal distribution

Mesokurtosis

data closely resembles that of a normal distribution

Platykurtosis

data points are more spread out, with fewer values clustered around the mean and more values in the tails

image:Bogleheads

Measures of Dispersion for Categorical Variables

The Variation Ratio


  • A measure of dispersion.

  • The proportion of cases which are not in the mode category.

  • The only measure of dispersion that can be used with categorical variables

The Variation Ratio

\[v = 1 - \frac{fm}{n}\]


\(fm\) = the frequency (number of cases) of the mode

\(n\) = sample size

Varition Ratio Example

We asked 1,984 individuals at the University what their favorite color was. We were left with four colors: red, orange, green, and blue. What is the variation score?


       colors
Red        43
Orange    211
Green     341
Blue     1389

\[v = 1 - \frac{fm}{n}\]

Recap

Dispersion


Kurtosis

  • Leptokurtosis

  • Mesokurtosis

  • Platykurtosis

The Variation Ratio

Measures of Dispersion for Continious Variables

Range


Variance


Standard Deviation

Range

A measure of the span of data.


A high range value indicates there is more dispersion, and the lower the range, the less the dispersion


\[Range = The \, Maximum \, Value - The \, Minimum \, Value\]

Range Example

Below is a data frame that contains the average salary within each district. Find the range.


          Salary
District1  87000
District2  91000
District3  66500
District4  98500
District5  96500
District6  97550
District7  97900
District8  97990

\[?\]

Variance

A measure of how spread the observed values are from the mean.

\[s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\]

\(s^2\) = sample variance
\(x_i\) = raw value
\(\bar{x}\) = mean values of all raw scores
\(n\) = number of observations

Break Down of Equation

  1. Find the \(\bar{x}\)

  2. Subtract the mean from each score \((x_i - \bar{x})\)

  3. Square the deviation score \((x_i - \bar{x})^2\)

  4. Add the squared deviations \({\sum(x_i - \bar{x})^2}\)

  5. Divide by \({n-1}\)

\[s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\]

Why n-1?

Suppose we draw n independent observations from a population (N), with a unknown population mean \((\mu)\) and unknown variance \((s^2)\).

Ideally we would use \((s^2)\) to find the average squared distance from the true mean,

\[s^2 = \frac{\sum(x_i - \mu)^2}{n}\]

Although we can’t! 😢

Because we don’t know our \((\mu)\).

We Sample it!

Since we don’t know \((\mu)\), we use our best estimate of it which is the sample mean \(\bar{x}\).

So let’s pull out the population variable \((\mu)\) and plug in our [sample variables (\(\bar{x}\)).

\[s^2 = \frac{\sum(x_i - \bar{x})^2}{n}\]

Although another small problems pops up!

The Problem (Population)


Population and the true \({\mu}\)


\[(-5,0)*--*-*-*----(0,0)--\stackrel{\mu}{|}--*--*--*--- (5,0)\]

The Problem (Sample)


Sample and the \(\bar{x}\)


\[(-5,0)*--*-*-----(0,0)--\stackrel{\mu}{|}--------- (5,0)\]


The sample mean can tend to underestimate or even overestimate the true \({\mu}\)

The Solution

We modify \(s^2 = \frac{\sum(x_i - \bar{x})^2}{n}\) to \(s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\)


And this provides an unbiased estimator of the population variance when using a sample.

image: khan academy

In Sum

When we are measuring a population, divide by n. (hint: There will be a \(\mu\) in the deviation score.)


When we are measuring a sample, divide by n-1. (hint: There will be a \(\bar{x}\) in the deviation score.)

Variance Example 1

1) Find the \(\bar{x}\)

2) Subtract the mean from each score \((x_i - \bar{x})\)

3) Square the deviation score \((x_i - \bar{x})^2\)

4) Add the squared deviations \({\sum(x_i - \bar{x})^2}\)

5) Divide by \({n-1}\)

   District Salary
1 District1    832
2 District2    931
3 District3   1468
4 District4   1021
5 District5   1039
6 District6   1515
7 District7   1138
8 District8    620

\[s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\]

Variance Example 2

1) Find the \(\bar{x}\)

2) Subtract the mean from each score \((x_i - \bar{x})\)

3) Square the deviation score \((x_i - \bar{x})^2\)

4) Add the squared deviations \({\sum(x_i - \bar{x})^2}\)

5) Divide by \({n-1}\)

   District Salary
1 District1    983
2 District2    993
3 District3   1047
4 District4   1002
5 District5   1004
6 District6   1051
7 District7   1014
8 District8    962

\[s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\]

Standard Deviation

A measure of how far observed values are from the mean.

It’s simply the square root of our variance!

\[\sigma=\sqrt{\frac{\sum_{} (x_{i} - \bar{x})^2}{n-1}}\] \(\sigma\) = standard deviation
\(x_i\) = raw value
\(\bar{x}\) = mean values of all raw scores
\(n\) = number of observations

Why square root it?

In our variance equation, we squared the sum of the deviation score \((x_i - \bar{x})\) to get rid of our negative values.

\[s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\] But by doing that we created an output that is nonsensical to our data and thus our interpretation.

\[\sigma=\sqrt{\frac{\sum_{} (x_{i} - \bar{x})^2}{n-1}}\]

So by squaring it, we normalize the sum of the deviation score, and thus the values make sense.

Standard Deviation Example

Take our calculation from variance example 1 and square it.

   District Salary
1 District1    832
2 District2    931
3 District3   1468
4 District4   1021
5 District5   1039
6 District6   1515
7 District7   1138
8 District8    620

\[\sigma=\sqrt{\frac{\sum_{} (x_{i} - \bar{x})^2}{n-1}}\]

Vizualize a Standard Deviation

Vizualize a Standard Deviation (Large)

Vizualize a Standard Deviation (Small)

The Normal Curve

On average, 68% of the sample will fall within 1 standard deviation of the mean, 95% at 2 standard deviations, and 99.7% will fall within 3 standard deviations.

image: medium

Have Wonderful Day!

Giphy